Rings Over Which Cyclics Are Direct Sums of Projective and Cs or Noetherian

Abstract

Abstract. R is called a right WV-ring if each simple right R-module is injectiverelative to proper cyclics. If R is a right WV-ring, then R is right uniform or a right Vring. It is shown that a right WV-ring R is right noetherian if and only if each right cyclic module is a direct sum of a projective module and a CS (complements are summands, a.k.a. 'extending modules') or noetherian module. For a finitely generated module M with projective socle over a V -ring R such that every subfactor of M is a direct sum of a projective module and a CS or noetherian module, we show M = X ⊕ T, where X is semisimple and T is noetherian with zero socle. In the case where M = R, we get R = S ⊕ T, where S is a semisimple artinian ring and T is a direct sum of right noetherian simple rings with zero socle. In addition, if R is a von Neumann regular ring, then it is semisimple artinian.2000 Mathematics Subject Classification. 16D50, 16D70, 16D80.

Some results on V-rings and weakly V-rings

“…In the last section of this note we give a condition for right WV-rings to be right Noetherian. The results obtained in this part significantly improve those in Section 3 of [18].…”

“…We would like to mention that our investigation in this paper is motivated by [18]. Furthermore, it is worth noting that Theorem 12 and Corollaries 13 and 14 significantly improve several results obtained in Section 3 of [18].…”

“…For the study of rings over which proper cyclic modules are injective or quasi-injective, we refer to [14,6,21,27]. Recently, using the concept of proper cyclic modules, Holston, Jain, and Leroy [18] defined a generalization of V-rings: A ring R is called a right weakly V-ring (briefly, right WV-ring) if every simple right R-module is injective relative to proper cyclic right R-modules. Left WV-rings are defined similarly.…”

“…So, in the noncommutative case, V-rings have become very interesting objects for researchers to investigate. 1 Other examples of insights into the structure of V-rings include the following: it was shown in [29] that the center Z (R) of a right V-ring R is a V-ring, and hence R becomes an algebra over the von Neumann regular ring Z (R); it was proved in [14, (7.36A)] that a semiprime right Goldie right V-ring is a direct sum of simple rings; and it was recently shown in [18], without any finiteness assumption, that a right V-domain is a simple ring. For more information on V-rings, see, e.g., [25,22,26,18,33,37].…”

“…Holston, Jain, and Leroy [18] introduced a generalization of V-rings using the so-called proper cyclic modules, and called these rings weakly V-rings. A cyclic right R-module is called proper cyclic if X R R R .…”